Answer
We can see that the limit does not exist.
Work Step by Step
Select different paths (on a hunch that there is no limit).
Along $y=0: \quad \lim _{x \rightarrow 0} \frac{0}{3 x^{2}}=\lim _{x \rightarrow 0} 0=0$
Along $y=x: \quad \lim _{x \rightarrow 0} \frac{x^{2}}{5 x^{2}}=\lim _{x \rightarrow 0} \frac{1}{5}=\frac{1}{5}$
We can see that the limit does not exist. After all, for the limit to exist, the limit must be the same regardless of the path chosen.
